Möbius ladder

Two views of the Möbius ladder M16. For an animation showing the transformation between the two views, see this file.

In graph theory, the Möbius ladderMn is a cubiccirculant graph with an even numbern of vertices, formed from an n-cycle by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is so-named because (with the exception of M6 = K3,3) Mn has exactly n/2 4-cycles[1] which link together by their shared edges to form a topological Möbius strip. Möbius ladders were named and first studied by Guy and Harary (1967).

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Every Möbius ladder is a nonplanarapex graph, meaning that it cannot be drawn without crossings in the plane but removing one vertex allows the remaining graph to be drawn without crossings. Möbius ladders have crossing number one, and can be embedded without crossings on a torus or projective plane. Thus, they are examples of toroidal graphs. Li (2005) explores embeddings of these graphs onto higher genus surfaces.

Möbius ladders are vertex-transitive – they have symmetries taking any vertex to any other vertex – but (again with the exception of M6) they are not edge-transitive. The edges from the cycle from which the ladder is formed can be distinguished from the rungs of the ladder, because each cycle edge belongs to a single 4-cycle, while each rung belongs to two such cycles. Therefore, there is no symmetry taking a cycle edge to a rung edge or vice versa.

The Möbius ladder M8 has 392 spanning trees; it and M6 have the most spanning trees among all cubic graphs with the same number of vertices.[2] However, the 10-vertex cubic graph with the most spanning trees is the Petersen graph, which is not a Möbius ladder.

Möbius ladders play an important role in the theory of graph minors. The earliest result of this type is a theorem of Klaus Wagner (1937) that graphs with no K5 minor can be formed by using clique-sum operations to combine planar graphs and the Möbius ladder M8; for this reason M8 is called the Wagner graph.

Gubser (1996) defines an almost-planar graph to be a nonplanar graph for which every nontrivial minor is planar; he shows that 3-connected almost-planar graphs are Möbius ladders or members of a small number of other families, and that other almost-planar graphs can be formed from these by a sequence of simple operations.

Maharry (2000) shows that almost all graphs that do not have a cube minor can be derived by a sequence of simple operations from Möbius ladders.

Walba, Richards & Haltiwanger (1982) first synthesized molecular structures in the form of a Möbius ladder, and since then this structure has been of interest in chemistry and chemical stereography,[4] especially in view of the ladder-like form of DNA molecules. With this application in mind, Flapan (1989) studies the mathematical symmetries of embeddings of Möbius ladders in R3.

Möbius ladders have also been used as the shape of a superconducting ring in experiments to study the effects of conductor topology on electron interactions.[5]